Smallest covering regions and highest density regions for discrete distributions

TL;DR

This paper introduces a novel iterative method for computing the smallest covering regions and highest density regions (HDRs) for arbitrary discrete distributions, addressing computational challenges and outperforming existing algorithms.

Contribution

It provides new theorems and an iterative algorithm for accurately computing smallest covering regions and HDRs in discrete distributions, a problem previously limited by computational complexity.

Findings

The proposed method accurately computes HDRs for any discrete distribution.

It outperforms existing algorithms in statistical software packages.

The method is demonstrated with pseudo-code and performance analysis.

Abstract

This paper examines the problem of computing a canonical smallest covering region for an arbitrary discrete probability distribution. This optimisation problem is similar to the classical 0-1 knapsack problem, but it involves optimisation over a set that may be countably infinite, raising a computational challenge that makes the problem non-trivial. To solve the problem we present theorems giving useful conditions for an optimising region and we develop an iterative one-at-a-time computational method to compute a canonical smallest covering region. We show how this can be programmed in pseudo-code and we examine the performance of our method. We compare this algorithm with other algorithms available in statistical computation packages to compute HDRs. We find that our method is the only one that accurately computes HDRs for arbitrary discrete distributions.